which \(H_{int}\)  is the interaction between system and environment For instance in the case which the long wavelength modes and short ones have the role of system and environment respectively, this interaction comes from the cubic terms in the perturbed action [1]}, [2]}. Moreover, these interaction terms can come from the extension of the standard model of particle physics or moduli of compactification in string theory.. Here the free Hamiltonian \(H_0\)  includes a kinetic term with Fourier transformation
\(H_{k}[S]= \frac{1}{2} \int \frac{d^{3} p}{(2\pi )^3} f_{s}(\tau ) \pi _{p}[S] \pi _{p}^{*}[S]\) 
